Abstract
In this paper, a new iterative scheme based on the extragradient-like method for finding a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a strongly positive linear bounded operator and the set of solutions of a mixed equilibrium problem is proposed. A strong convergence theorem for this iterative scheme in Hilbert spaces is established. Our results extend recent results announced by many others.MSC:49J30, 49J40, 47J25, 47H09.
Highlights
Let H be a real Hilbert space with the inner product ·, · and the norm ·
We denote by F(T) the set of fixed points of T
In this paper, motivated by Takahashi and Takahashi [ ], Ceng, Wang and Yao [ ], Peng and Yao [ ] and Qin, Shang and Su [ ], we introduce the general iterative scheme for finding a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of the generalized mixed equilibrium problem ( . ) and the set of solutions of the generalized equilibrium problem ( . ), which solves the variational inequality (A – γ f )x*, x – x* ≥, ∀x ∈ F, where F =
Summary
Let H be a real Hilbert space with the inner product ·, · and the norm ·. Peng and Yao [ ] considered iterative methods for finding a common element of the set of solutions of problem Qin et al [ ] studied the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a relaxed cocoercive mapping and the set of solutions of an equilibrium problem. Let f be a contraction of H into itself with a coefficient α ( < α < ) and let A be a strongly positive linear bounded operator with a coefficient γ > such that
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